Reasoning A b o u t Assumptions in Graphs of Models
Sanjaya A d d a n k i , R o b e r t o C r e m o n i n i , J . Scott P e n b e r t h y
I B M T . J . W a t s o n Research Center
P O Box 704
Y o r k t o w n Heights, N Y 10598
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Abstract
Solving design and analysis problems in physical worlds requires the representation of large
amounts of knowledge.
Recently, there has
been much interest in e x p l i c i t l y m a k i n g assumptions to decompose this knowledge i n t o
smaller Models. A crucial aspect of problemsolving paradigms based on models is t h a t
they include methods to a u t o m a t i c a l l y , and efficiently, select and change models. We represent physical domains as Graphs of Models,
where models are the nodes of the g r a p h and
the edges are the assumptions t h a t have to be
changed in going f r o m one model to the other.
T h i s paper describes the methods used in the
Graphs of Models p a r a d i g m for changing m o d els. T h i s knowledge can be represented qualit a t i v e l y , p e r m i t t i n g fast inference mechanisms
t h a t provide powerful model changing behaviors.
1
Introduction
An i m p o r t a n t aspect of real-world problem solving is
the a b i l i t y to represent, and reason w i t h , large amounts
of d o m a i n knowledge. For example, intelligent analysis
of the transmission shown in figure 1 requires thorough
knowledge of a b o u t three sophisticated t e x t b o o k s (e.g.
[ M a r t i n , 1982], [Arges and Palmer, 1963], [Boston Gear
Co., 1960]). U n f o r t u n a t e l y , declaratively representing
large amounts of knowledge leads to several problems;
e.g. many know ledge-base operations, such as inference
and checking consistency, are t y p i c a l l y exponential in the
size of the knowledge-base.
One approach to dealing w i t h the c o m p l e x i t y of large
scientific and engineering domains is to make assumptions on the w o r l d . M a k i n g assumptions p e r m i t s the
decomposition of large domains i n t o several smaller
knowledge-bases called models ( [ M u r t h y and A d d a n k i ,
1987], [Falkenhainer and Forbus, 1988]). A model may
be used in lieu of the larger d o m a i n knowledge-base if
its assumptions lead to an acceptable a p p r o x i m a t i o n of
the w o r l d .
T h e goal of the decomposition is to simplify problemsolving by p e r m i t t i n g analysis in the simplest model t h a t
is an acceptable a p p r o x i m a t i o n of the w o r l d . Ideally, the
1432
Knowledge Representation
appropriate model should be identified before the start
of analysis. However, in m a n y cases, the inadequacy of
a model becomes evident only t h r o u g h t r i a l and error.
Therefore, it is crucial t h a t problem-solving paradigms
based on models include mechanisms for a u t o m a t i c a l l y
and efficiently selecting a better model when analysis in
the current model is found to be in error.
In our p a r a d i g m , called Graphs of Models, models are
linked by directed edges. T h e edges specify how the
assumptions of the source model change in going to the
destination m o d e l . Figure 2 shows a p a r t of a simplistic
g r a p h of models for the d o m a i n of transmissions. Our
p a r a d i g m includes methods t h a t a u t o m a t i c a l l y change
models when the current model is inadequate. A brief
example w i l l help m o t i v a t e our methods.
Let us s t a r t an analysis of our transmission (figure 1)
using Model 1 of figure 2. Model 1 makes the assumptions t h a t there are no f r i c t i o n s in the w o r l d , t h a t all
o b j e c t s a r e r i g i d , t h a t e n e r g y i s c o n s e r v e d , and t h a t
all m a s s e s a r e u n i f o r m l y d i s t r i b u t e d . A s Model 1 i s
not an acceptable a p p r o x i m a t i o n of the w o r l d , it predicts
a r o t a t i o n a l acceleration value t h a t is higher t h a n the
e x p e r i m e n t a l l y observed value. T h i s error, or conflict,
invokes a reasoning process in w h i c h Model 1 first analyses the conflict to find t h a t the error is due to forces
t h a t have not been accounted for. N e x t , Model 1 uses
d o m a i n knowledge about its assumptions to find t h a t an
assumption change, o r t r a n s i t i o n , f r o m n o f r i c t i o n s t o
c o u l o m b f r i c t i o n s introduces new forces t h a t cause accelerations to decrease. F i n a l l y , M o d e l 1 finds the edge
o
m
models in the scientific or engineering sense of the term.
A model, in our paradigm, may make fifteen to twenty
assumptions and consists of two parts. The first per­
forms analysis within the model and the second deals
w i t h changing assumptions. A brief description of the
first part follows. The second part is the focus of this
paper.
Figure 3: A rule for the equation r = lα
t h a t best matches this assumption change, E d g e 1 , and
transfers control to the model at the end of this edge.
T h i s process requires four abilities. T h e ability to de­
tect conflicts, the a b i l i t y to determine how parameters
must change in order to eliminate conflicts, the a b i l i t y to
represent how assumption transitions affect parameters
in the w o r l d , and the a b i l i t y to use this knowledge in
selecting the next model.
We introduce the notions of delta-vectors to cap­
ture the q u a l i t a t i v e nature of parametric changes that
will eliminate conflicts 1 , and parameter-change rules to
capture domain-level knowledge about how assumption
transitions affect values of parameters.
A simple in­
ference mechanism uses delta-vectors and parameterchange rules to decide w h i c h assumptions to change. I n ­
t u i t i v e l y , delta-vectors may be seen as goals t h a t have to
be achieved, assumption transitions as the actions t h a t
w i l l satisfy these goals, and parameter-change rules as
the post-conditions of these actions. T h e mechanisms
are q u a l i t a t i v e in t h a t they are based on the (+ - 0) cal­
culus of [deKleer and B r o w n , 1984] and [Forbus, 1984].
T h e y have been used in four implementations of Graphs
of Models in the domains of Mechanics, T h e r m o d y n a m ­
ics, F l u i d s , and Geometric S t r u c t u r e .
Section 2 presents more details about Graphs of M o d ­
els. Section 3 describes conflicts and how we detect
thern. Section 4 describes delta-vectors and how they are
c o m p u t e d . Section 5 describes how parameter-change
rules represent knowledge about assumptions and Sec­
tion 6 describes the inference methods used for changing
assumptions. We end the paper w i t h a few of the many
questions t h a t we s t i l l have to address and the relation­
ship of our w o r k to others.
2
G r a p h s of M o d e l s
M a k i n g assumptions on the w o r l d permits organized re­
f o r m u l a t i o n s of d o m a i n knowledge t h a t are much like
1
For this paper we will assume that all conflicts arise from
incorrect values or expressions for physical parameters.
T h e analysis part of a model contains a knowledge­
base of rules in a first-order-like language. The rules are
exactly the rules of physics found in textbooks. T h e rule
in figure 3 describes the simple
law of rotational
dynamics;
is rotational acceleration,
is net-torque
and I is rotational inertia. The language is based on
a sorted calculus and the predicates are strongly t y p e d .
T h e sorts constitute the ontology of the world and are
organized into a hierarchy. The last parameter of every
predicate is a temporal variable t h a t refers to a time
point or interval over which the proposition is valid.
T h e temporal entities are supported by a powerful single
time line reasoner described in [Penberthy, 1988]. The
language supports expressions t h a t may be executed by
L I S P or passed to M A C S Y M A [ M a r t i n and Fateman, 197l]
for further evaluation; values of parameters are numeri­
cal values or symbolic expressions. Analysis consists of
n a t u r a l deduction t h a t is t i g h t l y controlled by modelspecific heuristics.
The principal advantage of a model is its specificity.
A model addresses a relatively narrow scope of phenom­
ena. T y p i c a l l y , models are much smaller than the entire
domain knowledge. It is relatively easy to represent the
knowledge that is valid w i t h i n a model. Further, the
specificity helps define very efficient model-specific anal­
ysis methods. Some apparent contradictions to these
claims t u r n out to be deceptive. For example, Maxwell's
equations and Navier-Stokes equations are very compact
and appear to define all the domain knowledge for elec­
trodynamics and fluid flow respectively.
However, it
is very hard to use either sets of equations, in practi­
cal analysis, w i t h o u t m a k i n g several assumptions on the
world.
Domain knowledge is represented as a G r a p h of M o d ­
els. The typical G r a p h may have tens of models. T h e
G r a p h of Models of a domain is complete; all models
have edges to all other models.
A typical edge has
around five assumption transitions. For each model, the
edges leading to other models are grouped i n t o prior­
i t y classes. T h e priorities allow a "distance" measure
in a complete graph by specifying the order in which
Addanki, Cremonini and Penberthy
1433
the edges are searched when changing models; "near"
edges are searched before "far" edges. The Graph of
Models of a domain is sparse; the graph does not con­
tain models for every combination of assumptions in the
domain. The models that are contained in a Graph of
Models are called "materialized models", other combina­
tions of assumptions are called "non-materialized mod­
els". It is obviously better to dynamically reconfigure
models based on the exact set of assumptions required.
Unfortunately, early work on reformulation of theories
shows that this is not easy to do ([Subramanian and
Genesereth, 1987], [Lowry, 1987]).
3
Conflicts
In our current approach conflicts can occur in three ways:
Empirically, Internallly, and through Inter-domain i n ­
teraction. An Empirical conflict is a mismatch between
the value of a parameter predicted by a model and the
value measured in the w o r l d . Empirical conflicts arise
in experimentally verifying a model's predictions; e.g.,
the conflict, described earlier, in the acceleration of the
transmission. Internal conflicts occur when a value de­
rived w i t h i n a model violates one of the model's as­
sumptions. For example, Model 1 makes the C o n s e r ­
v a t i v e S y s t e m s and the R i g i d B o d i e s assumptions.
If the R i g i d B o d i e s assumption is inappropriate, the
predicted values for energy will violate the C o n s e r v a ­
t i v e S y s t e m s assumption. Inter-Domain Conflicts oc­
cur when multiple domains interact in analysing a device
and two or more domains disagree over the value of a
common parameter [Addanki et ai , 1989].
Parameters can be numerical values or constraints.
Hence conflicts take one of the following forms:
Numerical-Numerical, where both values are numeri­
cal; Numerical-Constraint, where one value is numeri­
cal and doesn't satisfy the constraint; and ConstraintConstraint, where both values are constraints t h a t are
unequal.
Detecting empirical conflicts is a straightforward pro­
cess. At the end of the analysis session the system asks
the user to verify its predictions. The user gives the
system his/her set of values for the parameters and the
system compares its predictions against the user's mea­
surements. Detecting internal conflicts is a little more
difficult and requires knowledge about the assumptions
in the form of consistency rules (section 5.1). The detec­
tion of inter-domain conflicts is described in [Addanki et
ai , 1989].
4
Delta-Vectors
Delta-vectors represent the qualitative changes, to pa­
rameters values, that w i l l eliminate a conflict. A deltavector is an ordered pair; the first element is the name
of the parameter in conflict and the second element is a
qualitative vector representing the required change. This
follows our representation of vectors as a magnitude and
a direction u n i t vector. In our example the delta-vector
for the acceleration of the transmission is (α,(— (000)))
where the minus sign (—) signifies that the magnitude
of the acceleration is to be reduced and the (000) sig­
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Knowledge Representation
nifies the direction of the acceleration is correct. When
more than one parameter is found to be in conflict, the
delta-vectors are stored as a list called the delta-list.
4.1
Computing Delta-Vectors
Delta-vectors are first computed for the parameters ac­
tually in conflict. This delta-list is called the base-deltalist. T h e same mechanisms are used for computing deltavectors for empirical, internal, and inter-domain con­
flicts.
C o m p u t i n g delta-vectors in a Numerical-Numerical
conflict consists of computing the difference of the em­
pirical vector and the predicted vector and taking the
signs of the result. In our example if Model 1 predicts
(10.34 (000)) and α is measured to be (8.27 (000)),
the delta-vector is
There are two types of Numerical-Constraint conflicts.
In one the constraint is Causal; i.e. it is an equation or
inequality that is established by a model to compute a
parameter in terms of others. It is usually derived from
a physical law. In our example, Model 1 sets up the equa­
tion
to compute α. The other constraint is Im­
plicit, a physical principle t h a t must be satisfied by the
parameters of the model. Examples are the conservation
of energy and momentum in Mechanics and pv = nRT in
Thermodynamics. I m p l i c i t conflicts are usually internal
conflicts. Note t h a t the difference between causal and
implicit is in how a constraint is used by a model, and
is not inherent in the constraint.
Figure 4: Delta-vectors for Causal Constraints
If a predicted causal constraint fails to m a t c h a mea­
sured value, the model has already used the constraint to
compute its value for the parameter while detecting the
conflict. T h i s value is used to set up a vector-vector con­
flict w i t h the measured value (see figure 4). For example,
Model 1 predicts
=
Empirically
is (0.43 (000))
for
= (27.0 (000)), and / = 34.0. In detecting the
conflict Model 1 has already c o m p u t e d
= (0.79(000))
for
= ( 2 7 . 0 ( 0 0 0 ) ) and I = 34.0. Model 1 treats
as the predicted value and
as the measured value in
a vector-vector conflict. T h e resulting delta-vector is
Note t h a t delta-vectors say n o t h i n g about
how a constraint is to be changed; they only indicate how
the value of a parameter is to be changed.
T h e m e t h o d fails when the measured value is outside
the domain of the constraint; e.g. when the constraint
is
and the value measured is ( 5 , 5 ) . In
flict method w i t h the points of intersection being com­
puted by a stepwise sweep of a line passing through the
measured value. T h e resolution of the angular step is de­
creased iteratively u n t i l points of intersection are found.
T h e methods generalize to more dimensions. T h e dis­
cussion also assumed t h a t the model predicted a con­
straint t h a t was not satisfied by a measured value. T h e
same methods apply if the model predicts a value that
docs not satisfy a constraint measured in the w o r l d .
4.2
such situations the m o d e l exploits the fact t h a t deltavectors can only specify the q u a d r a n t 2 i n t o w h i c h , or the
axis along w h i c h , the change should be made; where the
quadrants are w i t h respect to the measured value. Deltavectors are c o m p u t e d by setting up artificial vectorvector conflicts between the measured value and a r t i f i ­
cial predicted values. T h e artificial predicted values are
generated by finding intersections of the lines t h a t pass
t h r o u g h the measured p o i n t , parallel to the axes, w i t h
the predicted constraint. Each p o i n t of intersection is an
artificial value. Intersections beyond a region pre-defined
by the user are ignored. If no intersections are f o u n d the
constraint lies entirely w i t h i n a q u a d r a n t and a single ar­
tificial p o i n t is generated by assigning values a r b i t r a r i l y
to the parameters of the constraint. In our example, the
lines y = 5 and x = 5 do not intersect the constraint.
Hence an a r b i t r a r y value, say ( 0 , 3 ) , on the constraint
is selected and is set up as a vector-vector conflict w i t h
( 5 , 5 ) . T h i s results in a delta-vector of (y, ( + ) ) .
Detecting constraint-constraint interactions, if the
constraints are linear, is p o l y n o m i a l ; if the constraints
are algebraic the p r o b l e m is exponentially space com­
plete; if the constraints include transcendentals the prob­
lem is u n c o m p u t a b l e . C u r r e n t l y our models handle the
very l i m i t e d classes of constraint-constraint conflicts in
which one constraint is s t r i c t l y greater t h a n the other.
However, m a n y constraints fall i n t o a few specific classes
for which special purpose routines can check for differ­
ences in order, coefficients, phase, etc. For example,
special purpose heuristics can resolve constraints on cur­
rents propagated t h r o u g h a circuit [Stallman and Sussm a n , 1977]. S i m i l a r l y , m a n y constraints on a single t r a n ­
scendental variable, e.g. xsin(0) < k, can be handled
w i t h a q u a d r a n t calculus t h a t assigns signs to the t r a n ­
scendental functions in each q u a d r a n t ; e.g., sin(θ) w i l l
be represented as
(++
) for the four quadrants. We
arc in the process of developing such algorithms.
In c o m p u t i n g delta-vectors for i m p l i c i t numericalconstraint conflicts the model directly generates a r t i f i ­
cial predicted values to compute the delta-vectors. T h e
first stage is o m m i t t e d due to the the lack of a causal
relationship.
T h e preceding discussion assumed t h a t the model gen­
erated only one constraint t h a t consisted of a simplyconnected p o r t i o n of space.
T h e methods generalize
easily to m u l t i p l e constraints if a simply-connected fea­
sible solution space exists.
Disconnected pieces, e.g.
xy — 1, are handled by the artificial vector-vector con­
2
Octants for 3-space.
Extending the Delta-List
T h e model also has to account for errors in intermedi­
ate parameters used in deriving the actual parameter
in conflict. For example, an error in r, an intermedi­
ate parameter, w i l l cause an erroneous value for a. T h e
base-delta-list is extended to include these parameters
and the final list is called the extendcd-dclta-list.
Figure 6: An Influence Net in Solid Dynamics
Our models m a i n t a i n a detailed proof graph of the
analysis. Backtracking through this graph finds all the
parameters t h a t affect the parameters in conflict. Each
model also explicitly represents k n o w n parameter depen­
dencies; e.g., force affects acceleration, t h a t in t u r n af­
fects velocity, etc. These dependencies are represented
as influences, [Forbus, 1984], in semantic nets called I n ­
fluence Nets (1-Nets). A sample 1-Net containing the i n ­
fluences between torque, acceleration, and other param­
eters in solid dynamics is seen in figure G. T h e required
change (or delta vector) of the parameter in conflict is
propagated t h r o u g h the I-net to determine the corre­
sponding changes in intermediate parameters.
I-nets
help finesse the problem of back-propagating a change
through complex equations.
In
our
example,
the
extended-delta-list
is:
where a was the
only parameter in the base-delta-list.
5
A s s u m p t i o n Knowledge
Assumptions affect the behaviors we expect to sec in the
w o r l d . For example, the coulomb frictions assumption
requires t h a t if t w o objects are in contact at a surface
a force prevents relative sliding of the objects, t h a t sys­
tems are dissipative, t h a t heat be generated, and it also
specifies the orders of magnitude of these effects. We rep­
resent t w o kinds of knowledge about assumptions, Con­
sistency Rules and Parameter-Change rules.
5.1
Consistency Rules
O f t e n , for an assumption to be v a l i d , the parameters
must satisfy given consistency constraints. For exam-
Addanki, Cremonini and Penberthy
1435
ple, i n M o d e l 1 , the c o n s e r v a t i v e s y s t e m s assumption
requires t h a t the energy w i t h i n the system being analyzed remain constant. T h e i d e a l gases a s s u m p t i o n
specifies a fixed constraint between the pressure, the
volume, the mass, and the t e m p e r a t u r e of gases in the
w o r l d . T h e l a m i n a r f l o w assumption i n f l u i d dynamics
requires t h a t the Reynolds n u m b e r of the flow remain
below 2300. These constraints are represented as consistency rules t h a t are checked periodically t h r o u g h o u t
analysis. An inconsistency is an i n t e r n a l conflict (discussed in section 3) and invokes the model changing apparatus.
introduces the p a r a m e t e r - c h a n g e predicate. T h e parameter change m a y be simple, as in a q u a l i t a t i v e (—)
or ( + ) , or complex, as i n any LlSP or M A C S Y M A statement in the compute predicate. T h e consequent may
i n t r o d u c e a parameter t h a t is not p a r t of the vocabulary
of the current m o d e l . For example, a simple no f r i c t i o n s model of dynamics m a y not include heat. Such
parameter-change rules are used for m a t c h i n g against
assertions f r o m other domains or f r o m the user. In our
experience, the situations in the antecedents can get
complex b u t the computations in the consequent stay
as simple as the inner p r o d u c t of vectors.
6
5.2
Parameter-Change Rules
Parameter-change rules represent the q u a l i t a t i v e effect
of assumption transitions on parameter values. T r a n s i tions t y p i c a l l y have four to six parameter-change rules.
T h e rule in figure 7 is taken f r o m the n o - f r i c t i o n s to
c o u l o m b - f r i c t i o n s t r a n s i t i o n . I t asserts t h a t i f t w o objects are in contact at a surface, there is an increase
in the component of the net-force along the surface of
contact. T h e antecedents of the rule describe the configuration of the objects and define f p to be the component of net-force parallel to the surface of contact. T h e
p a r a m e t e r - c h a n g e predicate in the consequent says t h a t
f p changes by a m o u n t
dvector1.
T h e compute predicate tells how to compute the a m o u n t of the change; the
a m o u n t of the change is ( + ) , a q u a l i t a t i v e increase in f p
under the t r a n s i t i o n . W h e n applied to gear t r a i n transmissions, this rule represents an overall decrease in the
effective force used to t r a n s m i t torques. A simpler rule,
useful in our example, states t h a t if t w o gears are meshed
there is a decrease in the m a g n i t u d e of the net-torque on
b o t h gears.
T h e antecedents of rules describe situations in which
parameters are affected by the assumption t r a n s i t i o n .
T h e language of the antecedents is exactly the same as
the d o m a i n language. T h e s i t u a t i o n is quite specific in
describing the conditions under w h i c h the parameterchange m a y occur; this p e r m i t s problem-specific selections of assumption transitions. In our example, the antecedents required objects to be in contact. Hence, the
t r a n s i t i o n can be rejected when the object of interest is
not in contact w i t h another object b u t a decrease in its
net-force is s t i l l required.
T h e consequents describe the parameters t h a t change
and how they change. T h e language of the consequents
1436
Knowledge Representation
Selecting a B e t t e r M o d e l
Recall t h a t the edges away f r o m a model are grouped
i n t o priority-classes. Selecting a better model consists
of stepping t h r o u g h these classes, searching each for an
acceptable edge. For each step, the delta-vectors specify
the desired changes in parameter values, the assumption transitions are actions t h a t realize these goals, and
the parameter-change rules specify how the transitions
affect parameter values. T h e final step, invoked if no
acceptable edge exists, determines the c o m b i n a t i o n of
assumption transitions t h a t best satisfies the delta-list;
this helps automate extending the G r a p h . An overview
of the a l g o r i t h m t h a t selects a model f r o m a given priori t y class is described in figure 8, T h e reader may find ituseful to refer to figure 8 for definitions of terms i n t r o duced in this section.
Figure 8: A s u m m a r y of our model selection a l g o r i t h m
T h e model starts w i t h the highest p r i o r i t y class of
edges. T h e first step is to find those assumption t r a n sitions t h a t satisfy at least one element of the deltalist. Each edge consists of a set of assumption transitions. Each t r a n s i t i o n has a set of associated parameterchange rules. T h e m o d e l collects all these parameterchange rules i n t o a t e m p o r a r y knowledge-base.
The
model then matches every delta-vector in the delta-list
against the rules in this knowledge base. W h e n the
consequent of a rule specifies a parameter change t h a t
matches a delta-vector, the m o d e l a t t e m p t s to match
the s i t u a t i o n in the antecedent of the rule against the
current description of the device.
If the antecedents
m a t c h i m m e d i a t e l y , the assumption t r a n s i t i o n associ­
ated w i t h this rule is placed in the C a n d i d a t e - T r a n s i t i o n Set ( C T S ) . 3 At the end of this backward-chaining the
C T S contains all the assumption transitions t h a t sat­
isfy at least one delta-vector. In our example, Model 1
has three edges leading away f r o m it and they are all
i n the same p r i o r i t y class. T w o transitions m a t c h ; n o
f r i c t i o n s t o c o u l o m b f r i c t i o n s matches
and u n i f o r m - m a s s - d i s t r i b u t i o n t o a r b i t r a r y - m a s s d i s t r i b u t i o n p a r t l y matches ( / , + ) .
T h e model m a i n t a i n s a transition-satisfaction-list
( T S L ) for each assumption t r a n s i t i o n in the C T S . T h e
T S L of a t r a n s i t i o n specifies how well the t r a n s i t i o n sat­
isfies each of the delta-vectors in the delta-list. A T S L
has as m a n y elements as the delta-list, has delta-vectors.
Each element is ( + ) , ( - ) , (0), or ( C ) . A ( + ) in the
n t h place indicates t h a t the t r a n s i t i o n satisfies the n t h
delta-vector, a ( —) t h a t it violates i t , a (0) t h a t it doesn't
affect i t , and a (C) t h a t it does change the parameter
but the direction of change is u n k n o w n . At this stage
the elements of the T S L s are either ( + ) s , ( C ) s or (0)s
because the backward-chaining finds all the positive ef­
fects of the transitions. T h e T S L s are also divided i n t o
t w o parts, one for the base and the other for the ex­
tended delta-lists. Recall t h a t the base-delta-list con­
tains α and the extended-delta-list contains and /. The
T S L for the f r i c t i o n s t r a n s i t i o n is
and the T S L for the m a s s - d i s t r i b u t i o n s transition is
T h e m o d e l then finds all the negative effects of the
transitions in the C T S by f o r w a r d - c h a i n i n g through their
parameter-change rules. T h i s is also a one step pro­
cess. These violations are stored as ( — )s in the T S L of
the transitions. O u r simple example does not include
any violations. T h e next step is to check if the I-Nets
show t h a t any non-(0) parameter in a T S L influences a
(0) parameter in the same T S L . Such an influence may
cause the (0) parameter to be satisfied or violated by
the t r a n s i t i o n . In our example, the I-Net in figure 3
shows t h a t
( + ) i n the f r i c t i o n s T S L , affects a , (0)
in the same T S L , p o s i t i v e l y ; hence the t r a n s i t i o n also
satisfies the delta-vector for a. T h e value for α in the
f r i c t i o n s T S L i s thus changed t o ( + ) and the T S L now
reads
S i m i l a r l y , since the I-Net
shows t h a t I affects α inversely, the T S L for the m a s s d i s t r i b u t i o n s T S L now reads
At
this p o i n t the T S L of each t r a n s i t i o n in the C T S provides
a composite p i c t u r e of how well the t r a n s i t i o n satisfies
the delta-list.
T h e next step is to compute a similar composite pic3
We require that the antecedents match immediately, with
no further backward-chaining. If the powerful problem solv­
ing mechanism in the model did not discover the situation in
the antecedent, it is unfair that the assumption reasoner be
asked to do so. It is clear that there will be cases where the
limitation will restrict the power of the system.
ture of how well each edge satisfies the delta-list. T h e
edge-satisfaction-list (ESL) of each edge in the highest
p r i o r i t y class is computed by vector a d d i t i o n of the T S L s
for each of the transitions in the edge; the rules of ad­
d i t i o n are the standard rules of the ( + , — , 0 ) calculus
w i t h (0) being the ambiguous condition (see [Forbus,
1984], [deKleer and B r o w n , 1984]). A ( C ) added to any­
t h i n g gives a ( C ) . Like the T S L s the ESLs have t w o
parts, one for the base- and the other for the extendeddelta-lists T h e ESLs for the three different edges out
of Model 1 are Edge 1 : { ( α \ + ) , ( r \ + , 7 \ C ) } , Edge 2 :
{ ( α \ 0 ) , ( r \ 0 , I \ 0 ) } , and Edge 3 :
{(α\C),
( r \ + , I\C)}
(see figure 1).
In evaluating the ESLs to select the next model, recall
t h a t the actual conflict involved only those parameters
in the base-delta-list. If the ESL for the base-delta-list
of any edge is perfect, i.e. all ( + ) s , the model at the end
of t h a t edge is placed in the Candidate Model Set ( C M S )
for consistency checking. If the C M S is non-empty, the
consistency rules of each assumption in each model in
the C M S are checked against k n o w n parameter values.
T h i s eliminates m o v i n g to models which may satisfy the
delta-list but are inconsistent w i t h the system being an­
alyzed. Finally, one of the remaining models in the C M S
is selected a r b i t r a r i l y . We do not have to look at edges
in the other p r i o r i t y groups because we cannot do better
than a perfect ESL t h a t meets all the consistency re­
quirements. If the C M S is empty, the TSLs are used in
repeating the process for edge groups of lower priorities.
In our example, the ESL for Edge1 satisfies i t . M o d e l 3 ,
at the end of Edge1 is placed in the C M S . The current
parameters satisfy the consistency rules of Models and
it is chosen as the next model.
If no existing edge provides a perfect m a t c h , the model
picks the next best match f r o m all its edges. ESLs are
evaluated numerically: ( + ) = +1, ( —) = — 1 , (C) = 0.5,
and a (0) = 0. A d d i n g the elements of an ESL results in a
numerical "goodness" value for the edge. 4 T h e models at
the end of the edges w i t h the highest goodness values are
placed in the C M S . T h e C M S is filtered for consistency,
and the next model is selected f r o m the filtered C M S .
Before control is transferred to the selected model the
model has the option of checking all combinations of as­
sumptions to see if a better match can be found. Recall
t h a t the Graph of Models is sparse and hence does not
include materialized models for every combination of as­
sumptions. Checking to see if a non-materialized model
results in a better match is useful for a u t o m a t i n g the
process of extending a Graph of Models. Unfortunately
the process is NP-coinplete; it is reducible to test-set
generation.
7
Conclusions and Future W o r k
Four implementations, albeit l i m i t e d to domains con­
taining four to eight models, in Geometric Structure,
T h e r m o d y n a m i c s , Mechanics, and Fluids, lead us to be­
lieve t h a t the Graphs of Models paradigm is a powerful
approach to representing complex, scientific and engi4
This is a very simplistic measure of "goodness" and we
are testing its limitations.
Addanki, Cremonini and Penberthy
1437
neering domains. M u c h of the power of the p a r a d i g m
comes f r o m its methods for a u t o m a t i c a l l y changing m o d els. T h e q u a l i t a t i v e mechanisms of delta-vectors and
parameter-change rules provide p o w e r f u l and efficient
m o d e l changing behaviors.
However, the mechanisms are l i m i t e d in m a n y ways.
One i m p o r t a n t l i m i t a t i o n has to do w i t h the p a u c i t y
of i n f o r m a t i o n in parameter-change rules. For example, humans t y p i c a l l y use m u c h Order of M a g n i t u d e i n f o r m a t i o n while evaluating the effects of changing ass u m p t i o n s . In our example, a h u m a n expert would compare the order of m a g n i t u d e of the change due to frictions w i t h the difference between the predicted value
and the measured value of the acceleration. T h e diffic u l t y w i t h using existing approaches to order of magn i t u d e reasoning, e.g. [ R a i m a n , 1986], [ M a v r o v o u n i o t i s
and Stephanopolous, 1987], or [ M u r t h y , 1988], is in sett i n g the threshold for the comparisons. These thresholds
are problem-dependent and have to be set d y n a m i c a l l y ,
an open p r o b l e m at this t i m e .
T h e mechanisms are also l i m i t e d by the simplistic nature of b u i l d i n g and c o m p a r i n g ESLs. Domains t y p i cally contain m u c h more i n f o r m a t i o n about the u t i l i t y
of models in a given s i t u a t i o n . T h i s i n f o r m a t i o n may be
in terms of tests t h a t can be carried out to check the
v a l i d i t y of a change, empirical likelihoods of assumption transitions, knowledge about parameter behaviors
as computed in the current m o d e l , and so o n . T h e current mechanisms make no use of any of this i n f o r m a t i o n .
A large part of our f u t u r e work w i l l focus on b u i l d i n g representation and inference mechanisms for h a n d l i n g these
types of knowledge.
T h e use of assumptions to simplify problem-solving
is not new. It is an inherent part of science and engineering. I n the A I l i t e r a t u r e , [deKleer and B r o w n ,
1984] emphasize the i m p o r t a n c e of m a k i n g m o d e l l i n g
assumptions explicit and acknowledge t h a t the problem of changing these assumptions is an i m p o r t a n t one.
[ M u r t h y and A d d a n k i , 1987] suggest the G r a p h of M o d els p a r a d i g m b u t present few details on its i n t e r n a l workings. [Falkenhainer and Forbus, 1988] re-emphasize the
importance of using assumptions to decompose complex
domains i n t o simpler models b u t do not indicate how
control is transferred f r o m one model to another. T h e oretical approaches such as [ L o w r y , 1987] and [Subramanian and Genesereth, 1987] have a t t e m p t e d to dynamically reconfigure theories based on the exact set of
assumptions required, b u t the results have been of l i m ited a p p l i c a b i l i t y to complex knowledge bases. For example, [Subramanian and Genesereth, 1987] require t h a t
the entire p r o b l e m be solved before deciding w h a t parts
of the knowledge base are irrelevant. F i n a l l y , in spite of
its l i m i t a t i o n s we believe t h a t a rigorous ( t h o u g h empirical) approach such as the G r a p h of Models paradigm
provides a powerful technique for representing physical
domains.
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